Proefschrift KWR

On Convolutional Processes and Dispersive Groundwater Flow, 1994


“While designing a bank groundwater recovery plant it was found that a temporary contamination of the river water would appear in the bank groundwater
recovered as a Gaussian-like function of time, no matter how the well field was arranged. An analysis of the flow pattern showed that the contaminated water
had to pass through a series of distinct flow systems. The relation between the in-going and the out-going signal of a single system could be given in the form
of a convolution integral. It was conjectured that the convolutional process was to be held responsible for the Gaussian-like shape of the output. In Chapter 2 of
this thesis this was found to be true under fairly general conditions: whenever a signal of finite duration has to pass through a long series oflinear systems, whose impulse responses fulfill certain conditions, then that signal will ultimately be transformed into the skew-Gaussian shape given by equation (2.45). The convolutional process turned out to be mathematically analogous to addition of random variables, and our result could be linked to the Central Limit Theorem
of mathematical statistics. Equation (2.45) is known, in that field of science, as Edgeworth’s asymptotic expansion of a random variable. The skew-Gaussian
shape of the output signal is characterized by a limited number of parameters, only two of whom (its mean or first moment and its variance or second central
moment) appear to increase in magnitude during the transport process. The other ones disappear gradually, as the signal moves along. The one to persist
longest is the skewness, which is related to the third central moment of the signal. These three moments suffice in many cases to describe an out-going signal
mathematically. It will often be possible to estimate their orders of magnitude without having to model the flow processes in all of the sub-systems in great

(Citaat: Maas, C. – On Convolutional Processes and Dispersive Groundwater Flow)

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